L(s) = 1 | + (−1.37 + 0.331i)2-s − 2.13·3-s + (1.78 − 0.910i)4-s + (2.93 − 0.707i)6-s + (1.19 + 2.35i)7-s + (−2.14 + 1.84i)8-s + 1.56·9-s + 2.33i·11-s + (−3.80 + 1.94i)12-s − 1.09i·13-s + (−2.42 − 2.84i)14-s + (2.34 − 3.24i)16-s − 4.98i·17-s + (−2.14 + 0.516i)18-s + 2.57·19-s + ⋯ |
L(s) = 1 | + (−0.972 + 0.234i)2-s − 1.23·3-s + (0.890 − 0.455i)4-s + (1.19 − 0.288i)6-s + (0.453 + 0.891i)7-s + (−0.759 + 0.650i)8-s + 0.520·9-s + 0.703i·11-s + (−1.09 + 0.561i)12-s − 0.302i·13-s + (−0.649 − 0.760i)14-s + (0.585 − 0.810i)16-s − 1.20i·17-s + (−0.506 + 0.121i)18-s + 0.590·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.107490 + 0.331159i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.107490 + 0.331159i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 - 0.331i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.19 - 2.35i)T \) |
good | 3 | \( 1 + 2.13T + 3T^{2} \) |
| 11 | \( 1 - 2.33iT - 11T^{2} \) |
| 13 | \( 1 + 1.09iT - 13T^{2} \) |
| 17 | \( 1 + 4.98iT - 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 6.04iT - 23T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 - 5.49T + 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 1.32iT - 43T^{2} \) |
| 47 | \( 1 + 9.74T + 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 0.620iT - 61T^{2} \) |
| 67 | \( 1 + 4.71iT - 67T^{2} \) |
| 71 | \( 1 - 11.9iT - 71T^{2} \) |
| 73 | \( 1 - 9.96iT - 73T^{2} \) |
| 79 | \( 1 - 10.6iT - 79T^{2} \) |
| 83 | \( 1 - 3.86T + 83T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98711709320459937246733057224, −9.707787101945686598021652821067, −9.369610738562379948221970619118, −8.100778634309668959315014640678, −7.34779511691237654409658668817, −6.38461866479730947800544211511, −5.48698629452584455551748222451, −4.95822568402514944858396537109, −2.86373269719429326903815721370, −1.42551319814252716825051515199,
0.31211304019048541208088032823, 1.61334657327578540262100042597, 3.39642492563641245931232026511, 4.62382228375135634507187568099, 5.91802007025431786189523943477, 6.54198234832540860006859115863, 7.55770576850110117609462840868, 8.357504198516034257681425001258, 9.341940475294420702396882420790, 10.52291488937517958072920496994